Revert "lists: Add list literal doc example."

[factor.git] / vm / bignum.cpp

// Copyright (C) 1989-94 Massachusetts Institute of Technology
// Portions copyright (C) 2004-2008 Slava Pestov

// This material was developed by the Scheme project at the Massachusetts
// Institute of Technology, Department of Electrical Engineering and
// Computer Science.  Permission to copy and modify this software, to
// redistribute either the original software or a modified version, and
// to use this software for any purpose is granted, subject to the
// following restrictions and understandings.

// 1. Any copy made of this software must include this copyright notice
// in full.

// 2. Users of this software agree to make their best efforts (a) to
// return to the MIT Scheme project any improvements or extensions that
// they make, so that these may be included in future releases; and (b)
// to inform MIT of noteworthy uses of this software.

// 3. All materials developed as a consequence of the use of this
// software shall duly acknowledge such use, in accordance with the usual
// standards of acknowledging credit in academic research.

// 4. MIT has made no warrantee or representation that the operation of
// this software will be error-free, and MIT is under no obligation to
// provide any services, by way of maintenance, update, or otherwise.

// 5. In conjunction with products arising from the use of this material,
// there shall be no use of the name of the Massachusetts Institute of
// Technology nor of any adaptation thereof in any advertising,
// promotional, or sales literature without prior written consent from
// MIT in each case.

// Changes for Scheme 48:
// *  - Converted to ANSI.
// *  - Added bitwise operations.
// *  - Added s48 to the beginning of all externally visible names.
// *  - Cached the bignum representations of -1, 0, and 1.

// Changes for Factor:
// *  - Adapt bignumint.h for Factor memory manager
// *  - Add more bignum <-> C type conversions
// *  - Remove unused functions
// *  - Add local variable GC root recording
// *  - Remove s48 prefix from function names
// *  - Various fixes for Win64
// *  - Port to C++
// *  - Added bignum_gcd implementation

#include "master.hpp"

namespace factor {

// Exports

int factor_vm::bignum_equal_p(bignum* x, bignum* y) {
  return ((BIGNUM_ZERO_P(x))
              ? (BIGNUM_ZERO_P(y))
              : ((!(BIGNUM_ZERO_P(y))) &&
                 ((BIGNUM_NEGATIVE_P(x)) ? (BIGNUM_NEGATIVE_P(y))
                                         : (!(BIGNUM_NEGATIVE_P(y)))) &&
                 (bignum_equal_p_unsigned(x, y))));
}

enum bignum_comparison factor_vm::bignum_compare(bignum* x, bignum* y) {
  return ((BIGNUM_ZERO_P(x)) ? ((BIGNUM_ZERO_P(y)) ? BIGNUM_COMPARISON_EQUAL
                                                   : (BIGNUM_NEGATIVE_P(y))
                                    ? BIGNUM_COMPARISON_GREATER
                                    : BIGNUM_COMPARISON_LESS)
                             : (BIGNUM_ZERO_P(y))
              ? ((BIGNUM_NEGATIVE_P(x)) ? BIGNUM_COMPARISON_LESS
                                        : BIGNUM_COMPARISON_GREATER)
              : (BIGNUM_NEGATIVE_P(x))
              ? ((BIGNUM_NEGATIVE_P(y)) ? (bignum_compare_unsigned(y, x))
                                        : (BIGNUM_COMPARISON_LESS))
              : ((BIGNUM_NEGATIVE_P(y)) ? (BIGNUM_COMPARISON_GREATER)
                                        : (bignum_compare_unsigned(x, y))));
}

// Allocates memory
bignum* factor_vm::bignum_add(bignum* x, bignum* y) {
  return (
      (BIGNUM_ZERO_P(x)) ? (y) : (BIGNUM_ZERO_P(y))
          ? (x)
          : ((BIGNUM_NEGATIVE_P(x))
                 ? ((BIGNUM_NEGATIVE_P(y)) ? (bignum_add_unsigned(x, y, 1))
                                           : (bignum_subtract_unsigned(y, x)))
                 : ((BIGNUM_NEGATIVE_P(y)) ? (bignum_subtract_unsigned(x, y))
                                           : (bignum_add_unsigned(x, y, 0)))));
}

// Allocates memory
bignum* factor_vm::bignum_subtract(bignum* x, bignum* y) {
  return ((BIGNUM_ZERO_P(x))
              ? ((BIGNUM_ZERO_P(y)) ? (y) : (bignum_new_sign(
                                                y, (!(BIGNUM_NEGATIVE_P(y))))))
              : ((BIGNUM_ZERO_P(y))
                     ? (x)
                     : ((BIGNUM_NEGATIVE_P(x))
                            ? ((BIGNUM_NEGATIVE_P(y))
                                   ? (bignum_subtract_unsigned(y, x))
                                   : (bignum_add_unsigned(x, y, 1)))
                            : ((BIGNUM_NEGATIVE_P(y))
                                   ? (bignum_add_unsigned(x, y, 0))
                                   : (bignum_subtract_unsigned(x, y))))));
}

#ifdef _WIN64
bignum *factor_vm::bignum_square(bignum* x_)
{
    return bignum_multiply(x_, x_);
}
#else
// Allocates memory
bignum *factor_vm::bignum_square(bignum* x_)
{
    data_root<bignum> x(x_, this);

    bignum_length_type length = (BIGNUM_LENGTH (x));
    bignum * z = (allot_bignum_zeroed ((length + length), 0));

    bignum_digit_type * scan_z = BIGNUM_START_PTR (z);
    bignum_digit_type * scan_x = BIGNUM_START_PTR (x);
    bignum_digit_type * end_x = scan_x + length;

    for (int i = 0; i < length; ++i) {
        bignum_twodigit_type carry;
        bignum_twodigit_type f = BIGNUM_REF (x, i);
        bignum_digit_type *pz = scan_z + (i << 1);
        bignum_digit_type *px = scan_x + i + 1;

        carry = *pz + f * f;
        *pz++ = carry & BIGNUM_DIGIT_MASK;
        carry >>= BIGNUM_DIGIT_LENGTH;
        BIGNUM_ASSERT (carry <= BIGNUM_DIGIT_MASK);

        f <<= 1;
        while (px < end_x)
        {
            carry += *pz + *px++ * f;
            *pz++ = carry & BIGNUM_DIGIT_MASK;
            carry >>= BIGNUM_DIGIT_LENGTH;
            BIGNUM_ASSERT (carry <= (BIGNUM_DIGIT_MASK << 1));
        }
        if (carry) {
            carry += *pz;
            *pz++ = carry & BIGNUM_DIGIT_MASK;
            carry >>= BIGNUM_DIGIT_LENGTH;
        }
        if (carry)
            *pz += carry & BIGNUM_DIGIT_MASK;
        BIGNUM_ASSERT ((carry >> BIGNUM_DIGIT_LENGTH) == 0);
    }
    return (bignum_trim (z));
}
#endif

// Allocates memory
bignum* factor_vm::bignum_multiply(bignum* x, bignum* y) {

#ifndef _WIN64
  if (x == y) {
    return bignum_square(x);
  }
#endif

  bignum_length_type x_length = (BIGNUM_LENGTH(x));
  bignum_length_type y_length = (BIGNUM_LENGTH(y));
  int negative_p = ((BIGNUM_NEGATIVE_P(x)) ? (!(BIGNUM_NEGATIVE_P(y)))
                                           : (BIGNUM_NEGATIVE_P(y)));
  if (BIGNUM_ZERO_P(x))
    return (x);
  if (BIGNUM_ZERO_P(y))
    return (y);
  if (x_length == 1) {
    bignum_digit_type digit = (BIGNUM_REF(x, 0));
    if (digit == 1)
      return (bignum_maybe_new_sign(y, negative_p));
    if (digit < BIGNUM_RADIX_ROOT)
      return (bignum_multiply_unsigned_small_factor(y, digit, negative_p));
  }
  if (y_length == 1) {
    bignum_digit_type digit = (BIGNUM_REF(y, 0));
    if (digit == 1)
      return (bignum_maybe_new_sign(x, negative_p));
    if (digit < BIGNUM_RADIX_ROOT)
      return (bignum_multiply_unsigned_small_factor(x, digit, negative_p));
  }
  return (bignum_multiply_unsigned(x, y, negative_p));
}

// Allocates memory
void factor_vm::bignum_divide(bignum* numerator, bignum* denominator,
                              bignum** quotient, bignum** remainder) {
  if (BIGNUM_ZERO_P(denominator)) {
    divide_by_zero_error();
    return;
  }
  if (BIGNUM_ZERO_P(numerator)) {
    (*quotient) = numerator;
    (*remainder) = numerator;
  } else {
    int r_negative_p = (BIGNUM_NEGATIVE_P(numerator));
    int q_negative_p =
        ((BIGNUM_NEGATIVE_P(denominator)) ? (!r_negative_p) : r_negative_p);
    switch (bignum_compare_unsigned(numerator, denominator)) {
      case BIGNUM_COMPARISON_EQUAL: {
        (*quotient) = (BIGNUM_ONE(q_negative_p));
        (*remainder) = (BIGNUM_ZERO());
        break;
      }
      case BIGNUM_COMPARISON_LESS: {
        (*quotient) = (BIGNUM_ZERO());
        (*remainder) = numerator;
        break;
      }
      case BIGNUM_COMPARISON_GREATER: {
        if ((BIGNUM_LENGTH(denominator)) == 1) {
          bignum_digit_type digit = (BIGNUM_REF(denominator, 0));
          if (digit == 1) {
            (*quotient) = (bignum_maybe_new_sign(numerator, q_negative_p));
            (*remainder) = (BIGNUM_ZERO());
            break;
          } else if (digit < BIGNUM_RADIX_ROOT) {
            bignum_divide_unsigned_small_denominator(numerator, digit, quotient,
                                                     remainder, q_negative_p,
                                                     r_negative_p);
            break;
          } else {
            bignum_divide_unsigned_medium_denominator(
                numerator, digit, quotient, remainder, q_negative_p,
                r_negative_p);
            break;
          }
        }
        bignum_divide_unsigned_large_denominator(
            numerator, denominator, quotient, remainder, q_negative_p,
            r_negative_p);
        break;
      }
    }
  }
}

// Allocates memory
bignum* factor_vm::bignum_quotient(bignum* numerator, bignum* denominator) {
  if (BIGNUM_ZERO_P(denominator)) {
    divide_by_zero_error();
    return (BIGNUM_OUT_OF_BAND);
  }
  if (BIGNUM_ZERO_P(numerator))
    return numerator;
  {
    int q_negative_p =
        ((BIGNUM_NEGATIVE_P(denominator)) ? (!(BIGNUM_NEGATIVE_P(numerator)))
                                          : (BIGNUM_NEGATIVE_P(numerator)));
    switch (bignum_compare_unsigned(numerator, denominator)) {
      case BIGNUM_COMPARISON_EQUAL:
        return (BIGNUM_ONE(q_negative_p));
      case BIGNUM_COMPARISON_LESS:
        return (BIGNUM_ZERO());
      case BIGNUM_COMPARISON_GREATER:
      default: // to appease gcc -Wall
               {
        bignum* quotient;
        if ((BIGNUM_LENGTH(denominator)) == 1) {
          bignum_digit_type digit = (BIGNUM_REF(denominator, 0));
          if (digit == 1)
            return (bignum_maybe_new_sign(numerator, q_negative_p));
          if (digit < BIGNUM_RADIX_ROOT)
            bignum_divide_unsigned_small_denominator(
                numerator, digit, (&quotient), ((bignum**)0), q_negative_p, 0);
          else
            bignum_divide_unsigned_medium_denominator(
                numerator, digit, (&quotient), ((bignum**)0), q_negative_p, 0);
        } else
          bignum_divide_unsigned_large_denominator(
              numerator, denominator, (&quotient), ((bignum**)0), q_negative_p,
              0);
        return (quotient);
      }
    }
  }
}

// Allocates memory
bignum* factor_vm::bignum_remainder(bignum* numerator, bignum* denominator) {
  if (BIGNUM_ZERO_P(denominator)) {
    divide_by_zero_error();
    return (BIGNUM_OUT_OF_BAND);
  }
  if (BIGNUM_ZERO_P(numerator))
    return numerator;
  switch (bignum_compare_unsigned(numerator, denominator)) {
    case BIGNUM_COMPARISON_EQUAL:
      return (BIGNUM_ZERO());
    case BIGNUM_COMPARISON_LESS:
      return numerator;
    case BIGNUM_COMPARISON_GREATER:
    default: // to appease gcc -Wall
             {
      bignum* remainder;
      if ((BIGNUM_LENGTH(denominator)) == 1) {
        bignum_digit_type digit = (BIGNUM_REF(denominator, 0));
        if (digit == 1)
          return (BIGNUM_ZERO());
        if (digit < BIGNUM_RADIX_ROOT)
          return (bignum_remainder_unsigned_small_denominator(
              numerator, digit, (BIGNUM_NEGATIVE_P(numerator))));
        bignum_divide_unsigned_medium_denominator(
            numerator, digit, ((bignum**)0), (&remainder), 0,
            (BIGNUM_NEGATIVE_P(numerator)));
      } else
        bignum_divide_unsigned_large_denominator(
            numerator, denominator, ((bignum**)0), (&remainder), 0,
            (BIGNUM_NEGATIVE_P(numerator)));
      return (remainder);
    }
  }
}

// cell_to_bignum, fixnum_to_bignum, long_long_to_bignum, ulong_long_to_bignum

// Allocates memory
#define FOO_TO_BIGNUM(name, type, stype, utype)                       \
  bignum* factor_vm::name##_to_bignum(type n) {                       \
    int negative_p;                                                   \
    /* Special cases win when these small constants are cached. */    \
    if (n == 0)                                                       \
      return (BIGNUM_ZERO());                                         \
    if (n == 1)                                                       \
      return (BIGNUM_ONE(0));                                         \
    if (n < (type) 0 && n == (type) - 1)                              \
      return (BIGNUM_ONE(1));                                         \
    {                                                                 \
      utype accumulator =                                             \
          ((negative_p = (n < (type) 0)) ? ((type)(-(stype) n)) : n); \
      if (accumulator < BIGNUM_RADIX)                                 \
      {                                                               \
        bignum* result = allot_bignum(1, negative_p);                 \
        bignum_digit_type* scan = (BIGNUM_START_PTR(result));         \
        *scan = (accumulator & BIGNUM_DIGIT_MASK);                    \
        return (result);                                              \
      } else {                                                        \
        bignum_digit_type result_digits[BIGNUM_DIGITS_FOR(type)];     \
        bignum_digit_type* end_digits = result_digits;                \
        do {                                                          \
          (*end_digits++) = (accumulator & BIGNUM_DIGIT_MASK);        \
          accumulator >>= BIGNUM_DIGIT_LENGTH;                        \
        } while (accumulator != 0);                                   \
        bignum* result =                                              \
           (allot_bignum((end_digits - result_digits), negative_p));  \
        bignum_digit_type* scan_digits = result_digits;               \
        bignum_digit_type* scan_result = (BIGNUM_START_PTR(result));  \
        while (scan_digits < end_digits)                              \
          (*scan_result++) = (*scan_digits++);                        \
        return (result);                                              \
      }                                                               \
    }                                                                 \
  }

FOO_TO_BIGNUM(cell, cell, fixnum, cell)
FOO_TO_BIGNUM(fixnum, fixnum, fixnum, cell)
FOO_TO_BIGNUM(long_long, int64_t, int64_t, uint64_t)
FOO_TO_BIGNUM(ulong_long, uint64_t, int64_t, uint64_t)

// cannot allocate memory
// bignum_to_cell, fixnum_to_cell, long_long_to_cell, ulong_long_to_cell
#define BIGNUM_TO_FOO(name, type, stype, utype)                            \
  type bignum_to_##name(bignum* bn) {                                      \
    if (BIGNUM_ZERO_P(bn))                                                 \
      return (0);                                                          \
    {                                                                      \
      utype accumulator = 0;                                               \
      bignum_digit_type* start = (BIGNUM_START_PTR(bn));                   \
      bignum_digit_type* scan = (start + (BIGNUM_LENGTH(bn)));             \
      while (start < scan)                                                 \
        accumulator = ((accumulator << BIGNUM_DIGIT_LENGTH) + (*--scan));  \
      return ((BIGNUM_NEGATIVE_P(bn)) ? ((type)(-(stype) accumulator))     \
                                      : accumulator);                      \
    }                                                                      \
  }

BIGNUM_TO_FOO(cell, cell, fixnum, cell)
BIGNUM_TO_FOO(fixnum, fixnum, fixnum, cell)
BIGNUM_TO_FOO(long_long, int64_t, int64_t, uint64_t)
BIGNUM_TO_FOO(ulong_long, uint64_t, int64_t, uint64_t)

bool bignum_fits_fixnum_p(bignum* bn) {
  fixnum len = BIGNUM_LENGTH(bn);
  if (len == 0)
    return true;
  if (len > 1)
    return false;
  bignum_digit_type dig = BIGNUM_START_PTR(bn)[0];
  return (BIGNUM_NEGATIVE_P(bn) && dig <= -fixnum_min) ||
      (!BIGNUM_NEGATIVE_P(bn) && dig <= fixnum_max);
}

cell bignum_maybe_to_fixnum(bignum* bn) {
  if (bignum_fits_fixnum_p(bn))
    return tag_fixnum(bignum_to_fixnum(bn));
  return tag<bignum>(bn);
}

// cannot allocate memory
fixnum factor_vm::bignum_to_fixnum_strict(bignum* bn) {

  if (!bignum_fits_fixnum_p(bn)) {
     general_error(ERROR_OUT_OF_FIXNUM_RANGE, tag<bignum>(bn), false_object);
  }
  fixnum fix = bignum_to_fixnum(bn);
  FACTOR_ASSERT(fix <= fixnum_max && fix >= fixnum_min);
  return fix;
}

#define DTB_WRITE_DIGIT(factor)                \
  {                                            \
    significand *= (factor);                   \
    digit = ((bignum_digit_type) significand); \
    (*--scan) = digit;                         \
    significand -= ((double)digit);            \
  }

#define inf std::numeric_limits<double>::infinity()

// Allocates memory
bignum* factor_vm::double_to_bignum(double x) {
  if (x == inf || x == -inf || x != x)
    return (BIGNUM_ZERO());
  int exponent;
  double significand = (frexp(x, (&exponent)));
  if (exponent <= 0)
    return (BIGNUM_ZERO());
  if (exponent == 1)
    return (BIGNUM_ONE(x < 0));
  if (significand < 0)
    significand = (-significand);
  {
    bignum_length_type length = (BIGNUM_BITS_TO_DIGITS(exponent));
    bignum* result = (allot_bignum(length, (x < 0)));
    bignum_digit_type* start = (BIGNUM_START_PTR(result));
    bignum_digit_type* scan = (start + length);
    bignum_digit_type digit;
    int odd_bits = (exponent % BIGNUM_DIGIT_LENGTH);
    if (odd_bits > 0)
      DTB_WRITE_DIGIT((fixnum)1 << odd_bits);
    while (start < scan) {
      if (significand == 0) {
        while (start < scan)
          (*--scan) = 0;
        break;
      }
      DTB_WRITE_DIGIT(BIGNUM_RADIX);
    }
    return (result);
  }
}

#undef DTB_WRITE_DIGIT

// Comparisons

int factor_vm::bignum_equal_p_unsigned(bignum* x, bignum* y) {
  bignum_length_type length = (BIGNUM_LENGTH(x));
  if (length != (BIGNUM_LENGTH(y)))
    return (0);
  else {
    bignum_digit_type* scan_x = (BIGNUM_START_PTR(x));
    bignum_digit_type* scan_y = (BIGNUM_START_PTR(y));
    bignum_digit_type* end_x = (scan_x + length);
    while (scan_x < end_x)
      if ((*scan_x++) != (*scan_y++))
        return (0);
    return (1);
  }
}

enum bignum_comparison factor_vm::bignum_compare_unsigned(bignum* x,
                                                          bignum* y) {
  bignum_length_type x_length = (BIGNUM_LENGTH(x));
  bignum_length_type y_length = (BIGNUM_LENGTH(y));
  if (x_length < y_length)
    return BIGNUM_COMPARISON_LESS;
  if (x_length > y_length)
    return BIGNUM_COMPARISON_GREATER;
  {
    bignum_digit_type* start_x = (BIGNUM_START_PTR(x));
    bignum_digit_type* scan_x = (start_x + x_length);
    bignum_digit_type* scan_y = ((BIGNUM_START_PTR(y)) + y_length);
    while (start_x < scan_x) {
      bignum_digit_type digit_x = (*--scan_x);
      bignum_digit_type digit_y = (*--scan_y);
      if (digit_x < digit_y)
        return BIGNUM_COMPARISON_LESS;
      if (digit_x > digit_y)
        return BIGNUM_COMPARISON_GREATER;
    }
  }
  return BIGNUM_COMPARISON_EQUAL;
}

// Addition

// Allocates memory
bignum* factor_vm::bignum_add_unsigned(bignum* x_, bignum* y_, int negative_p) {

  data_root<bignum> x(x_, this);
  data_root<bignum> y(y_, this);
  if ((BIGNUM_LENGTH(y)) > (BIGNUM_LENGTH(x))) {
    swap(x, y);
  }
  {
    bignum_length_type x_length = (BIGNUM_LENGTH(x));

    bignum* r = (allot_bignum((x_length + 1), negative_p));

    bignum_digit_type sum;
    bignum_digit_type carry = 0;
    bignum_digit_type* scan_x = (BIGNUM_START_PTR(x));
    bignum_digit_type* scan_r = (BIGNUM_START_PTR(r));
    {
      bignum_digit_type* scan_y = (BIGNUM_START_PTR(y));
      bignum_digit_type* end_y = (scan_y + (BIGNUM_LENGTH(y)));
      while (scan_y < end_y) {
        sum = ((*scan_x++) + (*scan_y++) + carry);
        if (sum < BIGNUM_RADIX) {
          (*scan_r++) = sum;
          carry = 0;
        } else {
          (*scan_r++) = (sum - BIGNUM_RADIX);
          carry = 1;
        }
      }
    }
    {
      bignum_digit_type* end_x = BIGNUM_START_PTR(x) + x_length;
      if (carry != 0)
        while (scan_x < end_x) {
          sum = ((*scan_x++) + 1);
          if (sum < BIGNUM_RADIX) {
            (*scan_r++) = sum;
            carry = 0;
            break;
          } else
            (*scan_r++) = (sum - BIGNUM_RADIX);
        }
      while (scan_x < end_x)
        (*scan_r++) = (*scan_x++);
    }
    if (carry != 0) {
      (*scan_r) = 1;
      return (r);
    }
    return (bignum_shorten_length(r, x_length));
  }
}

// Subtraction

// Allocates memory
bignum* factor_vm::bignum_subtract_unsigned(bignum* x_, bignum* y_) {

  data_root<bignum> x(x_, this);
  data_root<bignum> y(y_, this);

  int negative_p = 0;
  switch (bignum_compare_unsigned(x.untagged(), y.untagged())) {
    case BIGNUM_COMPARISON_EQUAL:
      return (BIGNUM_ZERO());
    case BIGNUM_COMPARISON_LESS:
      swap(x, y);
      negative_p = 1;
      break;
    case BIGNUM_COMPARISON_GREATER:
      negative_p = 0;
      break;
  }
  {
    bignum_length_type x_length = (BIGNUM_LENGTH(x));

    bignum* r = (allot_bignum(x_length, negative_p));

    bignum_digit_type difference;
    bignum_digit_type borrow = 0;
    bignum_digit_type* scan_x = BIGNUM_START_PTR(x);
    bignum_digit_type* scan_r = BIGNUM_START_PTR(r);
    {
      bignum_digit_type* scan_y = BIGNUM_START_PTR(y);
      bignum_digit_type* end_y = (scan_y + (BIGNUM_LENGTH(y)));
      while (scan_y < end_y) {
        difference = (((*scan_x++) - (*scan_y++)) - borrow);
        if (difference < 0) {
          (*scan_r++) = (difference + BIGNUM_RADIX);
          borrow = 1;
        } else {
          (*scan_r++) = difference;
          borrow = 0;
        }
      }
    }
    {
      bignum_digit_type* end_x = BIGNUM_START_PTR(x) + x_length;
      if (borrow != 0)
        while (scan_x < end_x) {
          difference = ((*scan_x++) - borrow);
          if (difference < 0)
            (*scan_r++) = (difference + BIGNUM_RADIX);
          else {
            (*scan_r++) = difference;
            borrow = 0;
            break;
          }
        }
      BIGNUM_ASSERT(borrow == 0);
      while (scan_x < end_x)
        (*scan_r++) = (*scan_x++);
    }
    return (bignum_trim(r));
  }
}

// Multiplication
// Maximum value for product_low or product_high:
// ((R * R) + (R * (R - 2)) + (R - 1))
// Maximum value for carry: ((R * (R - 1)) + (R - 1))
// where R == BIGNUM_RADIX_ROOT

// Allocates memory
bignum* factor_vm::bignum_multiply_unsigned(bignum* x_, bignum* y_,
                                            int negative_p) {

  data_root<bignum> x(x_, this);
  data_root<bignum> y(y_, this);

  if (BIGNUM_LENGTH(y) > BIGNUM_LENGTH(x)) {
    swap(x, y);
  }
  {
    bignum_digit_type carry;
    bignum_digit_type y_digit_low;
    bignum_digit_type y_digit_high;
    bignum_digit_type x_digit_low;
    bignum_digit_type x_digit_high;
    bignum_digit_type product_low;
    bignum_digit_type* scan_r;
    bignum_digit_type* scan_y;
    bignum_length_type x_length = BIGNUM_LENGTH(x);
    bignum_length_type y_length = BIGNUM_LENGTH(y);

    bignum* r = (allot_bignum_zeroed((x_length + y_length), negative_p));

    bignum_digit_type* scan_x = BIGNUM_START_PTR(x);
    bignum_digit_type* end_x = (scan_x + x_length);
    bignum_digit_type* start_y = BIGNUM_START_PTR(y);
    bignum_digit_type* end_y = (start_y + y_length);
    bignum_digit_type* start_r = (BIGNUM_START_PTR(r));
#define x_digit x_digit_high
#define y_digit y_digit_high
#define product_high carry
    while (scan_x < end_x) {
      x_digit = (*scan_x++);
      x_digit_low = (HD_LOW(x_digit));
      x_digit_high = (HD_HIGH(x_digit));
      carry = 0;
      scan_y = start_y;
      scan_r = (start_r++);
      while (scan_y < end_y) {
        y_digit = (*scan_y++);
        y_digit_low = (HD_LOW(y_digit));
        y_digit_high = (HD_HIGH(y_digit));
        product_low =
            ((*scan_r) + (x_digit_low * y_digit_low) + (HD_LOW(carry)));
        product_high =
            ((x_digit_high * y_digit_low) + (x_digit_low * y_digit_high) +
             (HD_HIGH(product_low)) + (HD_HIGH(carry)));
        (*scan_r++) = (HD_CONS((HD_LOW(product_high)), (HD_LOW(product_low))));
        carry = ((x_digit_high * y_digit_high) + (HD_HIGH(product_high)));
      }
      (*scan_r) += carry;
    }
    return (bignum_trim(r));
#undef x_digit
#undef y_digit
#undef product_high
  }
}

// Allocates memory
bignum* factor_vm::bignum_multiply_unsigned_small_factor(bignum* x_,
                                                         bignum_digit_type y,
                                                         int negative_p) {
  data_root<bignum> x(x_, this);

  bignum_length_type length_x = (BIGNUM_LENGTH(x));

  bignum* p = (allot_bignum((length_x + 1), negative_p));

  bignum_destructive_copy(x.untagged(), p);
  (BIGNUM_REF(p, length_x)) = 0;
  bignum_destructive_scale_up(p, y);
  return (bignum_trim(p));
}

void factor_vm::bignum_destructive_add(bignum* bn, bignum_digit_type n) {
  bignum_digit_type* scan = (BIGNUM_START_PTR(bn));
  bignum_digit_type digit;
  digit = ((*scan) + n);
  if (digit < BIGNUM_RADIX) {
    (*scan) = digit;
    return;
  }
  (*scan++) = (digit - BIGNUM_RADIX);
  while (1) {
    digit = ((*scan) + 1);
    if (digit < BIGNUM_RADIX) {
      (*scan) = digit;
      return;
    }
    (*scan++) = (digit - BIGNUM_RADIX);
  }
}

void factor_vm::bignum_destructive_scale_up(bignum* bn,
                                            bignum_digit_type factor) {
  bignum_digit_type carry = 0;
  bignum_digit_type* scan = (BIGNUM_START_PTR(bn));
  bignum_digit_type two_digits;
  bignum_digit_type product_low;
#define product_high carry
  bignum_digit_type* end = (scan + (BIGNUM_LENGTH(bn)));
  BIGNUM_ASSERT((factor > 1) && (factor < BIGNUM_RADIX_ROOT));
  while (scan < end) {
    two_digits = (*scan);
    product_low = ((factor * (HD_LOW(two_digits))) + (HD_LOW(carry)));
    product_high = ((factor * (HD_HIGH(two_digits))) + (HD_HIGH(product_low)) +
                    (HD_HIGH(carry)));
    (*scan++) = (HD_CONS((HD_LOW(product_high)), (HD_LOW(product_low))));
    carry = (HD_HIGH(product_high));
  }
  // A carry here would be an overflow, i.e. it would not fit.
  // Hopefully the callers allocate enough space that this will
  // never happen.
  BIGNUM_ASSERT(carry == 0);
  return;
#undef product_high
}

// Division

// For help understanding this algorithm, see:
// Knuth, Donald E., "The Art of Computer Programming",
// volume 2, "Seminumerical Algorithms"
// section 4.3.1, "Multiple-Precision Arithmetic".

// Allocates memory
void factor_vm::bignum_divide_unsigned_large_denominator(
    bignum* numerator_, bignum* denominator_,
    bignum** quotient, bignum** remainder,
    int q_negative_p, int r_negative_p) {

  data_root<bignum> numerator(numerator_, this);
  data_root<bignum> denominator(denominator_, this);

  bignum_length_type length_n = BIGNUM_LENGTH(numerator) + 1;
  bignum_length_type length_d = BIGNUM_LENGTH(denominator);

  data_root<bignum> u(allot_bignum(length_n, r_negative_p), this);

  int shift = 0;
  BIGNUM_ASSERT(length_d > 1);
  {
    bignum_digit_type v1 = BIGNUM_REF(denominator.untagged(), length_d - 1);
    while (v1 < (BIGNUM_RADIX / 2)) {
      v1 <<= 1;
      shift += 1;
    }
  }

  if (quotient != NULL) {
    bignum *q_ = allot_bignum(length_n - length_d, q_negative_p);
    data_root<bignum> q(q_, this);

    if (shift == 0) {
      bignum_destructive_copy(numerator.untagged(), u.untagged());
      (BIGNUM_REF(u.untagged(), (length_n - 1))) = 0;
      bignum_divide_unsigned_normalized(u.untagged(),
                                        denominator.untagged(),
                                        q.untagged());
    } else {
      bignum* v = allot_bignum(length_d, 0);
      bignum_destructive_normalization(numerator.untagged(),
                                       u.untagged(),
                                       shift);
      bignum_destructive_normalization(denominator.untagged(), v, shift);
      bignum_divide_unsigned_normalized(u.untagged(), v, q.untagged());
      if (remainder != NULL)
        bignum_destructive_unnormalization(u.untagged(), shift);
    }

    q.set_untagged(bignum_trim(q.untagged()));
    *quotient = q.untagged();
  } else {

    if (shift == 0) {
        bignum_destructive_copy(numerator.untagged(), u.untagged());
        (BIGNUM_REF(u.untagged(), (length_n - 1))) = 0;
        bignum_divide_unsigned_normalized(u.untagged(),
                                          denominator.untagged(),
                                          NULL);
      } else {
        bignum* v = allot_bignum(length_d, 0);
        bignum_destructive_normalization(numerator.untagged(),
                                         u.untagged(),
                                         shift);
        bignum_destructive_normalization(denominator.untagged(),
                                         v,
                                         shift);
        bignum_divide_unsigned_normalized(u.untagged(), v, NULL);
        if (remainder != NULL)
          bignum_destructive_unnormalization(u.untagged(), shift);
      }
  }

  u.set_untagged(bignum_trim(u.untagged()));
  if (remainder != NULL)
    *remainder = u.untagged();
}

void factor_vm::bignum_divide_unsigned_normalized(bignum* u, bignum* v,
                                                  bignum* q) {
  bignum_length_type u_length = (BIGNUM_LENGTH(u));
  bignum_length_type v_length = (BIGNUM_LENGTH(v));
  bignum_digit_type* u_start = (BIGNUM_START_PTR(u));
  bignum_digit_type* u_scan = (u_start + u_length);
  bignum_digit_type* u_scan_limit = (u_start + v_length);
  bignum_digit_type* u_scan_start = (u_scan - v_length);
  bignum_digit_type* v_start = (BIGNUM_START_PTR(v));
  bignum_digit_type* v_end = (v_start + v_length);
  bignum_digit_type* q_scan = NULL;
  bignum_digit_type v1 = (v_end[-1]);
  bignum_digit_type v2 = (v_end[-2]);
  bignum_digit_type ph; // high half of double-digit product
  bignum_digit_type pl; // low half of double-digit product
  bignum_digit_type guess;
  bignum_digit_type gh; // high half-digit of guess
  bignum_digit_type ch; // high half of double-digit comparand
  bignum_digit_type v2l = (HD_LOW(v2));
  bignum_digit_type v2h = (HD_HIGH(v2));
  bignum_digit_type cl; // low half of double-digit comparand
#define gl ph           // low half-digit of guess
#define uj pl
#define qj ph
  bignum_digit_type gm; // memory loc for reference parameter
  if (q != BIGNUM_OUT_OF_BAND)
    q_scan = ((BIGNUM_START_PTR(q)) + (BIGNUM_LENGTH(q)));
  while (u_scan_limit < u_scan) {
    uj = (*--u_scan);
    if (uj != v1) {
      // comparand =
      // (((((uj * BIGNUM_RADIX) + uj1) % v1) * BIGNUM_RADIX) + uj2);
      // guess = (((uj * BIGNUM_RADIX) + uj1) / v1);
      cl = (u_scan[-2]);
      ch = (bignum_digit_divide(uj, (u_scan[-1]), v1, (&gm)));
      guess = gm;
    } else {
      cl = (u_scan[-2]);
      ch = ((u_scan[-1]) + v1);
      guess = (BIGNUM_RADIX - 1);
    }
    while (1) {
      // product = (guess * v2);
      gl = (HD_LOW(guess));
      gh = (HD_HIGH(guess));
      pl = (v2l * gl);
      ph = ((v2l * gh) + (v2h * gl) + (HD_HIGH(pl)));
      pl = (HD_CONS((HD_LOW(ph)), (HD_LOW(pl))));
      ph = ((v2h * gh) + (HD_HIGH(ph)));
      // if (comparand >= product)
      if ((ch > ph) || ((ch == ph) && (cl >= pl)))
        break;
      guess -= 1;
      // comparand += (v1 << BIGNUM_DIGIT_LENGTH)
      ch += v1;
      // if (comparand >= (BIGNUM_RADIX * BIGNUM_RADIX))
      if (ch >= BIGNUM_RADIX)
        break;
    }
    qj = (bignum_divide_subtract(v_start, v_end, guess, (--u_scan_start)));
    if (q != BIGNUM_OUT_OF_BAND)
      (*--q_scan) = qj;
  }
  return;
#undef gl
#undef uj
#undef qj
}

bignum_digit_type factor_vm::bignum_divide_subtract(
    bignum_digit_type* v_start, bignum_digit_type* v_end,
    bignum_digit_type guess, bignum_digit_type* u_start) {
  bignum_digit_type* v_scan = v_start;
  bignum_digit_type* u_scan = u_start;
  bignum_digit_type carry = 0;
  if (guess == 0)
    return (0);
  {
    bignum_digit_type gl = (HD_LOW(guess));
    bignum_digit_type gh = (HD_HIGH(guess));
    bignum_digit_type v;
    bignum_digit_type pl;
    bignum_digit_type vl;
#define vh v
#define ph carry
#define diff pl
    while (v_scan < v_end) {
      v = (*v_scan++);
      vl = (HD_LOW(v));
      vh = (HD_HIGH(v));
      pl = ((vl * gl) + (HD_LOW(carry)));
      ph = ((vl * gh) + (vh * gl) + (HD_HIGH(pl)) + (HD_HIGH(carry)));
      diff = ((*u_scan) - (HD_CONS((HD_LOW(ph)), (HD_LOW(pl)))));
      if (diff < 0) {
        (*u_scan++) = (diff + BIGNUM_RADIX);
        carry = ((vh * gh) + (HD_HIGH(ph)) + 1);
      } else {
        (*u_scan++) = diff;
        carry = ((vh * gh) + (HD_HIGH(ph)));
      }
    }
    if (carry == 0)
      return (guess);
    diff = ((*u_scan) - carry);
    if (diff < 0)
      (*u_scan) = (diff + BIGNUM_RADIX);
    else {
      (*u_scan) = diff;
      return (guess);
    }
#undef vh
#undef ph
#undef diff
  }
  // Subtraction generated carry, implying guess is one too large.
  // Add v back in to bring it back down.
  v_scan = v_start;
  u_scan = u_start;
  carry = 0;
  while (v_scan < v_end) {
    bignum_digit_type sum = ((*v_scan++) + (*u_scan) + carry);
    if (sum < BIGNUM_RADIX) {
      (*u_scan++) = sum;
      carry = 0;
    } else {
      (*u_scan++) = (sum - BIGNUM_RADIX);
      carry = 1;
    }
  }
  if (carry == 1) {
    bignum_digit_type sum = ((*u_scan) + carry);
    (*u_scan) = ((sum < BIGNUM_RADIX) ? sum : (sum - BIGNUM_RADIX));
  }
  return (guess - 1);
}

// Allocates memory
void factor_vm::bignum_divide_unsigned_medium_denominator(
    bignum* numerator_, bignum_digit_type denominator, bignum** quotient,
    bignum** remainder, int q_negative_p, int r_negative_p) {

  data_root<bignum> numerator(numerator_, this);

  bignum_length_type length_n = (BIGNUM_LENGTH(numerator));

  int shift = 0;
  // Because `bignum_digit_divide' requires a normalized denominator.
  while (denominator < (BIGNUM_RADIX / 2)) {
    denominator <<= 1;
    shift += 1;
  }

  bignum_length_type length_q = (shift == 0) ? length_n : length_n + 1;
  data_root<bignum> q(allot_bignum(length_q, q_negative_p), this);
  if (shift == 0) {
    bignum_destructive_copy(numerator.untagged(), q.untagged());
  } else {
    bignum_destructive_normalization(numerator.untagged(), q.untagged(), shift);
  }
  {
    bignum_digit_type r = 0;
    bignum_digit_type* start = (BIGNUM_START_PTR(q));
    bignum_digit_type* scan = (start + length_q);
    bignum_digit_type qj;

    while (start < scan) {
      r = (bignum_digit_divide(r, (*--scan), denominator, (&qj)));
      (*scan) = qj;
    }

    q.set_untagged(bignum_trim(q.untagged()));

    if (remainder != ((bignum**)0)) {
      if (shift != 0)
        r >>= shift;

      (*remainder) = (bignum_digit_to_bignum(r, r_negative_p));
    }

    if (quotient != ((bignum**)0))
      (*quotient) = q.untagged();
  }
  return;
}

void factor_vm::bignum_destructive_normalization(bignum* source, bignum* target,
                                                 int shift_left) {
  bignum_digit_type digit;
  bignum_digit_type* scan_source = (BIGNUM_START_PTR(source));
  bignum_digit_type carry = 0;
  bignum_digit_type* scan_target = (BIGNUM_START_PTR(target));
  bignum_digit_type* end_source = (scan_source + (BIGNUM_LENGTH(source)));
  bignum_digit_type* end_target = (scan_target + (BIGNUM_LENGTH(target)));
  int shift_right = (BIGNUM_DIGIT_LENGTH - shift_left);
  bignum_digit_type mask = (((cell)1 << shift_right) - 1);
  while (scan_source < end_source) {
    digit = (*scan_source++);
    (*scan_target++) = (((digit & mask) << shift_left) | carry);
    carry = (digit >> shift_right);
  }
  if (scan_target < end_target)
    (*scan_target) = carry;
  else
    BIGNUM_ASSERT(carry == 0);
  return;
}

void factor_vm::bignum_destructive_unnormalization(bignum* bn,
                                                   int shift_right) {
  bignum_digit_type* start = (BIGNUM_START_PTR(bn));
  bignum_digit_type* scan = (start + (BIGNUM_LENGTH(bn)));
  bignum_digit_type digit;
  bignum_digit_type carry = 0;
  int shift_left = (BIGNUM_DIGIT_LENGTH - shift_right);
  bignum_digit_type mask = (((fixnum)1 << shift_right) - 1);
  while (start < scan) {
    digit = (*--scan);
    (*scan) = ((digit >> shift_right) | carry);
    carry = ((digit & mask) << shift_left);
  }
  BIGNUM_ASSERT(carry == 0);
  return;
}

// This is a reduced version of the division algorithm, applied to the
// case of dividing two bignum digits by one bignum digit. It is
// assumed that the numerator, denominator are normalized.

#define BDD_STEP(qn, j)                                          \
  {                                                              \
    uj = (u[j]);                                                 \
    if (uj != v1) {                                              \
      uj_uj1 = (HD_CONS(uj, (u[j + 1])));                        \
      guess = (uj_uj1 / v1);                                     \
      comparand = (HD_CONS((uj_uj1 % v1), (u[j + 2])));          \
    } else {                                                     \
      guess = (BIGNUM_RADIX_ROOT - 1);                           \
      comparand = (HD_CONS(((u[j + 1]) + v1), (u[j + 2])));      \
    }                                                            \
    while ((guess * v2) > comparand) {                           \
      guess -= 1;                                                \
      comparand += (v1 << BIGNUM_HALF_DIGIT_LENGTH);             \
      if (comparand >= BIGNUM_RADIX)                             \
        break;                                                   \
    }                                                            \
    qn = (bignum_digit_divide_subtract(v1, v2, guess, (&u[j]))); \
  }

bignum_digit_type factor_vm::bignum_digit_divide(
    bignum_digit_type uh, bignum_digit_type ul, bignum_digit_type v,
    bignum_digit_type* q) // return value
    {
  bignum_digit_type guess;
  bignum_digit_type comparand;
  bignum_digit_type v1 = (HD_HIGH(v));
  bignum_digit_type v2 = (HD_LOW(v));
  bignum_digit_type uj;
  bignum_digit_type uj_uj1;
  bignum_digit_type q1;
  bignum_digit_type q2;
  bignum_digit_type u[4];
  if (uh == 0) {
    if (ul < v) {
      (*q) = 0;
      return (ul);
    } else if (ul == v) {
      (*q) = 1;
      return (0);
    }
  }
  (u[0]) = (HD_HIGH(uh));
  (u[1]) = (HD_LOW(uh));
  (u[2]) = (HD_HIGH(ul));
  (u[3]) = (HD_LOW(ul));
  v1 = (HD_HIGH(v));
  v2 = (HD_LOW(v));
  BDD_STEP(q1, 0);
  BDD_STEP(q2, 1);
  (*q) = (HD_CONS(q1, q2));
  return (HD_CONS((u[2]), (u[3])));
}

#undef BDD_STEP

#define BDDS_MULSUB(vn, un, carry_in)    \
  {                                      \
    product = ((vn * guess) + carry_in); \
    diff = (un - (HD_LOW(product)));     \
    if (diff < 0) {                      \
      un = (diff + BIGNUM_RADIX_ROOT);   \
      carry = ((HD_HIGH(product)) + 1);  \
    } else {                             \
      un = diff;                         \
      carry = (HD_HIGH(product));        \
    }                                    \
  }

#define BDDS_ADD(vn, un, carry_in)    \
  {                                   \
    sum = (vn + un + carry_in);       \
    if (sum < BIGNUM_RADIX_ROOT) {    \
      un = sum;                       \
      carry = 0;                      \
    } else {                          \
      un = (sum - BIGNUM_RADIX_ROOT); \
      carry = 1;                      \
    }                                 \
  }

bignum_digit_type factor_vm::bignum_digit_divide_subtract(
    bignum_digit_type v1, bignum_digit_type v2, bignum_digit_type guess,
    bignum_digit_type* u) {
  {
    bignum_digit_type product;
    bignum_digit_type diff;
    bignum_digit_type carry;
    BDDS_MULSUB(v2, (u[2]), 0);
    BDDS_MULSUB(v1, (u[1]), carry);
    if (carry == 0)
      return (guess);
    diff = ((u[0]) - carry);
    if (diff < 0)
      (u[0]) = (diff + BIGNUM_RADIX);
    else {
      (u[0]) = diff;
      return (guess);
    }
  }
  {
    bignum_digit_type sum;
    bignum_digit_type carry;
    BDDS_ADD(v2, (u[2]), 0);
    BDDS_ADD(v1, (u[1]), carry);
    if (carry == 1)
      (u[0]) += 1;
  }
  return (guess - 1);
}

#undef BDDS_MULSUB
#undef BDDS_ADD

// Allocates memory
void factor_vm::bignum_divide_unsigned_small_denominator(
    bignum* numerator_, bignum_digit_type denominator, bignum** quotient,
    bignum** remainder, int q_negative_p, int r_negative_p) {
  data_root<bignum> numerator(numerator_, this);

  bignum* q_ = bignum_new_sign(numerator.untagged(), q_negative_p);
  data_root<bignum> q(q_, this);

  bignum_digit_type r = bignum_destructive_scale_down(q.untagged(), denominator);

  q.set_untagged(bignum_trim(q.untagged()));

  if (remainder != ((bignum**)0))
    (*remainder) = bignum_digit_to_bignum(r, r_negative_p);

  (*quotient) = q.untagged();

  return;
}

// Given (denominator > 1), it is fairly easy to show that
// (quotient_high < BIGNUM_RADIX_ROOT), after which it is easy to see
// that all digits are < BIGNUM_RADIX.

bignum_digit_type factor_vm::bignum_destructive_scale_down(
    bignum* bn, bignum_digit_type denominator) {
  bignum_digit_type numerator;
  bignum_digit_type remainder = 0;
  bignum_digit_type two_digits;
#define quotient_high remainder
  bignum_digit_type* start = (BIGNUM_START_PTR(bn));
  bignum_digit_type* scan = (start + (BIGNUM_LENGTH(bn)));
  BIGNUM_ASSERT((denominator > 1) && (denominator < BIGNUM_RADIX_ROOT));
  while (start < scan) {
    two_digits = (*--scan);
    numerator = (HD_CONS(remainder, (HD_HIGH(two_digits))));
    quotient_high = (numerator / denominator);
    numerator = (HD_CONS((numerator % denominator), (HD_LOW(two_digits))));
    (*scan) = (HD_CONS(quotient_high, (numerator / denominator)));
    remainder = (numerator % denominator);
  }
  return (remainder);
#undef quotient_high
}

// Allocates memory
bignum* factor_vm::bignum_remainder_unsigned_small_denominator(
    bignum* n, bignum_digit_type d, int negative_p) {
  bignum_digit_type two_digits;
  bignum_digit_type* start = (BIGNUM_START_PTR(n));
  bignum_digit_type* scan = (start + (BIGNUM_LENGTH(n)));
  bignum_digit_type r = 0;
  BIGNUM_ASSERT((d > 1) && (d < BIGNUM_RADIX_ROOT));
  while (start < scan) {
    two_digits = (*--scan);
    r = ((HD_CONS(((HD_CONS(r, (HD_HIGH(two_digits)))) % d),
                  (HD_LOW(two_digits)))) %
         d);
  }
  return (bignum_digit_to_bignum(r, negative_p));
}

// Allocates memory
bignum* factor_vm::bignum_digit_to_bignum(bignum_digit_type digit,
                                          int negative_p) {
  if (digit == 0)
    return (BIGNUM_ZERO());
  else {
    bignum* result = (allot_bignum(1, negative_p));
    (BIGNUM_REF(result, 0)) = digit;
    return (result);
  }
}

// Allocates memory
bignum* factor_vm::allot_bignum(bignum_length_type length, int negative_p) {
  BIGNUM_ASSERT((length >= 0) || (length < BIGNUM_RADIX));
  bignum* result = allot_uninitialized_array<bignum>(length + 1);
  BIGNUM_SET_NEGATIVE_P(result, negative_p);
  return (result);
}

// Allocates memory
bignum* factor_vm::allot_bignum_zeroed(bignum_length_type length,
                                       int negative_p) {
  bignum* result = allot_bignum(length, negative_p);
  bignum_digit_type* scan = (BIGNUM_START_PTR(result));
  bignum_digit_type* end = (scan + length);
  while (scan < end)
    (*scan++) = 0;
  return (result);
}

// Allocates memory
bignum* factor_vm::bignum_shorten_length(bignum* bn,
                                         bignum_length_type length) {
  bignum_length_type current_length = (BIGNUM_LENGTH(bn));
  BIGNUM_ASSERT((length >= 0) || (length <= current_length));
  if (length < current_length) {
    bn = reallot_array(bn, length + 1);
    BIGNUM_SET_NEGATIVE_P(bn, (length != 0) && (BIGNUM_NEGATIVE_P(bn)));
  }
  return (bn);
}

// Allocates memory
bignum* factor_vm::bignum_trim(bignum* bn) {
  bignum_digit_type* start = (BIGNUM_START_PTR(bn));
  bignum_digit_type* end = (start + (BIGNUM_LENGTH(bn)));
  bignum_digit_type* scan = end;
  while ((start <= scan) && ((*--scan) == 0))
    ;
  scan += 1;
  if (scan < end) {
    bignum_length_type length = (scan - start);
    bn = reallot_array(bn, length + 1);
    BIGNUM_SET_NEGATIVE_P(bn, (length != 0) && (BIGNUM_NEGATIVE_P(bn)));
  }
  return (bn);
}

// Copying

// Allocates memory
bignum* factor_vm::bignum_new_sign(bignum* x_, int negative_p) {
  data_root<bignum> x(x_, this);
  bignum* result = allot_bignum(BIGNUM_LENGTH(x), negative_p);
  bignum_destructive_copy(x.untagged(), result);
  return result;
}

// Allocates memory
bignum* factor_vm::bignum_maybe_new_sign(bignum* x_, int negative_p) {
  if ((BIGNUM_NEGATIVE_P(x_)) ? negative_p : (!negative_p))
    return x_;
  else {
    return bignum_new_sign(x_, negative_p);
  }
}

void factor_vm::bignum_destructive_copy(bignum* source, bignum* target) {
  bignum_digit_type* scan_source = (BIGNUM_START_PTR(source));
  bignum_digit_type* end_source = (scan_source + (BIGNUM_LENGTH(source)));
  bignum_digit_type* scan_target = (BIGNUM_START_PTR(target));
  while (scan_source < end_source)
    (*scan_target++) = (*scan_source++);
  return;
}

// * Added bitwise operations (and oddp).

// Allocates memory
bignum* factor_vm::bignum_bitwise_not(bignum* x_) {

  int carry = 1;
  bignum_length_type size = BIGNUM_LENGTH(x_);
  int is_negative = BIGNUM_NEGATIVE_P(x_);
  data_root<bignum> x(x_, this);
  data_root<bignum> y(allot_bignum(size, is_negative ? 0 : 1), this);

  bignum_digit_type* scan_x = BIGNUM_START_PTR(x);
  bignum_digit_type* end_x = scan_x + size;
  bignum_digit_type* scan_y = BIGNUM_START_PTR(y);

  if (is_negative) {
    while (scan_x < end_x) {
      if (*scan_x == 0) {
        *scan_y++ = BIGNUM_RADIX - 1;
        scan_x++;
      } else {
        *scan_y++ = *scan_x++ - 1;
        carry = 0;
        break;
      }
    }
  } else {
    while (scan_x < end_x) {
      if (*scan_x == (BIGNUM_RADIX - 1)) {
        *scan_y++ = 0;
        scan_x++;
      } else {
        *scan_y++ = *scan_x++ + 1;
        carry = 0;
        break;
      }
    }
  }

  while (scan_x < end_x) {
    *scan_y++ = *scan_x++;
  }

  if (carry) {
    bignum* ret = allot_bignum(size + 1, BIGNUM_NEGATIVE_P(y));
    bignum_destructive_copy(y.untagged(), ret);
    bignum_digit_type* ret_start = BIGNUM_START_PTR(ret);
    *(ret_start + size) = 1;
    return ret;
  } else {
    return bignum_trim(y.untagged());
  }
}

// Allocates memory
bignum* factor_vm::bignum_arithmetic_shift(bignum* arg1, fixnum n) {
  if (BIGNUM_NEGATIVE_P(arg1) && n < 0)
    return bignum_bitwise_not(
        bignum_magnitude_ash(bignum_bitwise_not(arg1), n));
  else
    return bignum_magnitude_ash(arg1, n);
}

#define AND_OP 0
#define IOR_OP 1
#define XOR_OP 2

// Allocates memory
bignum* factor_vm::bignum_bitwise_and(bignum* arg1, bignum* arg2) {
  return ((BIGNUM_NEGATIVE_P(arg1)) ? (BIGNUM_NEGATIVE_P(arg2))
              ? bignum_negneg_bitwise_op(AND_OP, arg1, arg2)
              : bignum_posneg_bitwise_op(AND_OP, arg2, arg1)
              : (BIGNUM_NEGATIVE_P(arg2))
              ? bignum_posneg_bitwise_op(AND_OP, arg1, arg2)
              : bignum_pospos_bitwise_op(AND_OP, arg1, arg2));
}

// Allocates memory
bignum* factor_vm::bignum_bitwise_ior(bignum* arg1, bignum* arg2) {
  return ((BIGNUM_NEGATIVE_P(arg1)) ? (BIGNUM_NEGATIVE_P(arg2))
              ? bignum_negneg_bitwise_op(IOR_OP, arg1, arg2)
              : bignum_posneg_bitwise_op(IOR_OP, arg2, arg1)
              : (BIGNUM_NEGATIVE_P(arg2))
              ? bignum_posneg_bitwise_op(IOR_OP, arg1, arg2)
              : bignum_pospos_bitwise_op(IOR_OP, arg1, arg2));
}

// Allocates memory
bignum* factor_vm::bignum_bitwise_xor(bignum* arg1, bignum* arg2) {
  return ((BIGNUM_NEGATIVE_P(arg1)) ? (BIGNUM_NEGATIVE_P(arg2))
              ? bignum_negneg_bitwise_op(XOR_OP, arg1, arg2)
              : bignum_posneg_bitwise_op(XOR_OP, arg2, arg1)
              : (BIGNUM_NEGATIVE_P(arg2))
              ? bignum_posneg_bitwise_op(XOR_OP, arg1, arg2)
              : bignum_pospos_bitwise_op(XOR_OP, arg1, arg2));
}

// Allocates memory
// ash for the magnitude
// assume arg1 is a big number, n is a long
bignum* factor_vm::bignum_magnitude_ash(bignum* arg1_, fixnum n) {

  data_root<bignum> arg1(arg1_, this);

  bignum* result = NULL;
  bignum_digit_type* scan1;
  bignum_digit_type* scanr;
  bignum_digit_type* end;

  fixnum digit_offset, bit_offset;

  if (BIGNUM_ZERO_P(arg1))
    return arg1.untagged();

  if (n > 0) {
    digit_offset = n / BIGNUM_DIGIT_LENGTH;
    bit_offset = n % BIGNUM_DIGIT_LENGTH;

    result = allot_bignum_zeroed(BIGNUM_LENGTH(arg1) + digit_offset + 1,
                                 BIGNUM_NEGATIVE_P(arg1));

    scanr = BIGNUM_START_PTR(result) + digit_offset;
    scan1 = BIGNUM_START_PTR(arg1);
    end = scan1 + BIGNUM_LENGTH(arg1);

    while (scan1 < end) {
      *scanr = *scanr | (*scan1 & BIGNUM_DIGIT_MASK) << bit_offset;
      *scanr = *scanr & BIGNUM_DIGIT_MASK;
      scanr++;
      *scanr = *scan1++ >> (BIGNUM_DIGIT_LENGTH - bit_offset);
      *scanr = *scanr & BIGNUM_DIGIT_MASK;
    }
  } else if (n < 0 && (-n >= (BIGNUM_LENGTH(arg1) * (bignum_length_type)
                              BIGNUM_DIGIT_LENGTH))) {
    result = BIGNUM_ZERO();
  } else if (n < 0) {
    digit_offset = -n / BIGNUM_DIGIT_LENGTH;
    bit_offset = -n % BIGNUM_DIGIT_LENGTH;

    result = allot_bignum_zeroed(BIGNUM_LENGTH(arg1) - digit_offset,
                                 BIGNUM_NEGATIVE_P(arg1));

    scanr = BIGNUM_START_PTR(result);
    scan1 = BIGNUM_START_PTR(arg1) + digit_offset;
    end = scanr + BIGNUM_LENGTH(result) - 1;

    while (scanr < end) {
      *scanr = (*scan1++ & BIGNUM_DIGIT_MASK) >> bit_offset;
      *scanr = (*scanr | *scan1 << (BIGNUM_DIGIT_LENGTH - bit_offset)) &
               BIGNUM_DIGIT_MASK;
      scanr++;
    }
    *scanr = (*scan1++ & BIGNUM_DIGIT_MASK) >> bit_offset;
  } else if (n == 0) {
    result = arg1.untagged();
  }

  return bignum_trim(result);
}

// Allocates memory
bignum* factor_vm::bignum_pospos_bitwise_op(int op, bignum* arg1_,
                                            bignum* arg2_) {
  data_root<bignum> arg1(arg1_, this);
  data_root<bignum> arg2(arg2_, this);

  bignum_length_type max_length;

  bignum_digit_type* scan1, *end1, digit1;
  bignum_digit_type* scan2, *end2, digit2;
  bignum_digit_type* scanr, *endr;

  max_length =
      (BIGNUM_LENGTH(arg1) > BIGNUM_LENGTH(arg2)) ? BIGNUM_LENGTH(arg1)
                                                  : BIGNUM_LENGTH(arg2);

  bignum* result = allot_bignum(max_length, 0);

  scanr = BIGNUM_START_PTR(result);
  scan1 = BIGNUM_START_PTR(arg1);
  scan2 = BIGNUM_START_PTR(arg2);
  endr = scanr + max_length;
  end1 = scan1 + BIGNUM_LENGTH(arg1);
  end2 = scan2 + BIGNUM_LENGTH(arg2);

  while (scanr < endr) {
    digit1 = (scan1 < end1) ? *scan1++ : 0;
    digit2 = (scan2 < end2) ? *scan2++ : 0;
    *scanr++ =
        (op == AND_OP) ? digit1 & digit2 : (op == IOR_OP) ? digit1 | digit2
                                                          : digit1 ^ digit2;
  }
  return bignum_trim(result);
}

// Allocates memory
bignum* factor_vm::bignum_posneg_bitwise_op(int op, bignum* arg1_,
                                            bignum* arg2_) {
  data_root<bignum> arg1(arg1_, this);
  data_root<bignum> arg2(arg2_, this);

  bignum_length_type max_length;

  bignum_digit_type* scan1, *end1, digit1;
  bignum_digit_type* scan2, *end2, digit2, carry2;
  bignum_digit_type* scanr, *endr;

  char neg_p = op == IOR_OP || op == XOR_OP;

  max_length =
      (BIGNUM_LENGTH(arg1) > BIGNUM_LENGTH(arg2) + 1) ? BIGNUM_LENGTH(arg1)
                                                      : BIGNUM_LENGTH(arg2) + 1;

  bignum* result = allot_bignum(max_length, neg_p);

  scanr = BIGNUM_START_PTR(result);
  scan1 = BIGNUM_START_PTR(arg1);
  scan2 = BIGNUM_START_PTR(arg2);
  endr = scanr + max_length;
  end1 = scan1 + BIGNUM_LENGTH(arg1);
  end2 = scan2 + BIGNUM_LENGTH(arg2);

  carry2 = 1;

  while (scanr < endr) {
    digit1 = (scan1 < end1) ? *scan1++ : 0;
    digit2 = (~((scan2 < end2) ? *scan2++ : 0) & BIGNUM_DIGIT_MASK) + carry2;

    if (digit2 < BIGNUM_RADIX)
      carry2 = 0;
    else {
      digit2 = (digit2 - BIGNUM_RADIX);
      carry2 = 1;
    }

    *scanr++ =
        (op == AND_OP) ? digit1 & digit2 : (op == IOR_OP) ? digit1 | digit2
                                                          : digit1 ^ digit2;
  }

  if (neg_p)
    bignum_negate_magnitude(result);

  return bignum_trim(result);
}

// Allocates memory
bignum* factor_vm::bignum_negneg_bitwise_op(int op, bignum* arg1_,
                                            bignum* arg2_) {
  data_root<bignum> arg1(arg1_, this);
  data_root<bignum> arg2(arg2_, this);

  bignum_length_type max_length;

  bignum_digit_type* scan1, *end1, digit1, carry1;
  bignum_digit_type* scan2, *end2, digit2, carry2;
  bignum_digit_type* scanr, *endr;

  char neg_p = op == AND_OP || op == IOR_OP;

  max_length =
      (BIGNUM_LENGTH(arg1) > BIGNUM_LENGTH(arg2)) ? BIGNUM_LENGTH(arg1) + 1
                                                  : BIGNUM_LENGTH(arg2) + 1;

  bignum* result = allot_bignum(max_length, neg_p);

  scanr = BIGNUM_START_PTR(result);
  scan1 = BIGNUM_START_PTR(arg1);
  scan2 = BIGNUM_START_PTR(arg2);
  endr = scanr + max_length;
  end1 = scan1 + BIGNUM_LENGTH(arg1);
  end2 = scan2 + BIGNUM_LENGTH(arg2);

  carry1 = 1;
  carry2 = 1;

  while (scanr < endr) {
    digit1 = (~((scan1 < end1) ? *scan1++ : 0) & BIGNUM_DIGIT_MASK) + carry1;
    digit2 = (~((scan2 < end2) ? *scan2++ : 0) & BIGNUM_DIGIT_MASK) + carry2;

    if (digit1 < BIGNUM_RADIX)
      carry1 = 0;
    else {
      digit1 = (digit1 - BIGNUM_RADIX);
      carry1 = 1;
    }

    if (digit2 < BIGNUM_RADIX)
      carry2 = 0;
    else {
      digit2 = (digit2 - BIGNUM_RADIX);
      carry2 = 1;
    }

    *scanr++ =
        (op == AND_OP) ? digit1 & digit2 : (op == IOR_OP) ? digit1 | digit2
                                                          : digit1 ^ digit2;
  }

  if (neg_p)
    bignum_negate_magnitude(result);

  return bignum_trim(result);
}

void factor_vm::bignum_negate_magnitude(bignum* arg) {
  bignum_digit_type* scan;
  bignum_digit_type* end;
  bignum_digit_type digit;
  bignum_digit_type carry;

  scan = BIGNUM_START_PTR(arg);
  end = scan + BIGNUM_LENGTH(arg);

  carry = 1;

  while (scan < end) {
    digit = (~ * scan & BIGNUM_DIGIT_MASK) + carry;

    if (digit < BIGNUM_RADIX)
      carry = 0;
    else {
      digit = (digit - BIGNUM_RADIX);
      carry = 1;
    }

    *scan++ = digit;
  }
}

// Allocates memory
bignum* factor_vm::bignum_integer_length(bignum* x_) {
  data_root<bignum> x(x_, this);
  bignum_length_type index = ((BIGNUM_LENGTH(x)) - 1);
  bignum_digit_type digit = (BIGNUM_REF(x, index));
  bignum_digit_type carry = 0;
  bignum* result;

  while (digit > 1) {
    carry += 1;
    digit >>= 1;
  }

  if (index < BIGNUM_RADIX_ROOT) {
     result = allot_bignum(1, 0);
     (BIGNUM_REF(result, 0)) = (index * BIGNUM_DIGIT_LENGTH) + carry;
  } else {
     result = allot_bignum(2, 0);
     (BIGNUM_REF(result, 0)) = index;
     (BIGNUM_REF(result, 1)) = 0;
     bignum_destructive_scale_up(result, BIGNUM_DIGIT_LENGTH);
     bignum_destructive_add(result, carry);
  }
  return (bignum_trim(result));
}

// Allocates memory
int factor_vm::bignum_logbitp(int shift, bignum* arg) {
  return ((BIGNUM_NEGATIVE_P(arg))
              ? !bignum_unsigned_logbitp(shift, bignum_bitwise_not(arg))
              : bignum_unsigned_logbitp(shift, arg));
}

int factor_vm::bignum_unsigned_logbitp(int shift, bignum* bn) {
  bignum_length_type len = (BIGNUM_LENGTH(bn));
  int index = shift / BIGNUM_DIGIT_LENGTH;
  if (index >= len)
    return 0;
  bignum_digit_type digit = (BIGNUM_REF(bn, index));
  int p = shift % BIGNUM_DIGIT_LENGTH;
  bignum_digit_type mask = ((fixnum)1) << p;
  return (digit & mask) ? 1 : 0;
}

#ifdef _WIN64
// Allocates memory.
bignum* factor_vm::bignum_gcd(bignum* a_, bignum* b_) {

  data_root<bignum> a(a_, this);
  data_root<bignum> b(b_, this);

  // Copies of a and b with that are both positive.
  data_root<bignum> ac(bignum_maybe_new_sign(a.untagged(), 0), this);
  data_root<bignum> bc(bignum_maybe_new_sign(b.untagged(), 0), this);

  if (bignum_compare(ac.untagged(), bc.untagged()) == BIGNUM_COMPARISON_LESS) {
    swap(ac, bc);
  }

  while (BIGNUM_LENGTH(bc) != 0) {
    data_root<bignum> d(bignum_remainder(ac.untagged(), bc.untagged()), this);
    if (d.untagged() == BIGNUM_OUT_OF_BAND) {
      return d.untagged();
    }
    ac = bc;
    bc = d;
  }
  return ac.untagged();
}
#else
// Allocates memory
bignum* factor_vm::bignum_gcd(bignum* a_, bignum* b_) {
  data_root<bignum> a(a_, this);
  data_root<bignum> b(b_, this);
  bignum_twodigit_type x, y, q, s, t, A, B, C, D;
  int nbits, k;
  bignum_length_type size_a, size_b, size_c;
  bignum_digit_type* scan_a, *scan_b, *scan_c, *scan_d;
  bignum_digit_type* a_end, *b_end, *c_end;

  // clone the bignums so we can modify them in-place
  size_a = BIGNUM_LENGTH(a);
  data_root<bignum> c(allot_bignum(size_a, 0), this);
  // c = allot_bignum(size_a, 0);
  scan_a = BIGNUM_START_PTR(a);
  a_end = scan_a + size_a;
  scan_c = BIGNUM_START_PTR(c);
  while (scan_a < a_end)
    (*scan_c++) = (*scan_a++);
  a = c;
  size_b = BIGNUM_LENGTH(b);
  data_root<bignum> d(allot_bignum(size_b, 0), this);
  scan_b = BIGNUM_START_PTR(b);
  b_end = scan_b + size_b;
  scan_d = BIGNUM_START_PTR(d);
  while (scan_b < b_end)
    (*scan_d++) = (*scan_b++);
  b = d;

  // Initial reduction: make sure that 0 <= b <= a.
  if (bignum_compare(a.untagged(), b.untagged()) == BIGNUM_COMPARISON_LESS) {
    swap(a, b);
    std::swap(size_a, size_b);
  }

  while (size_a > 1) {
    nbits = log2(BIGNUM_REF(a, size_a - 1));
    x = ((BIGNUM_REF(a, size_a - 1) << (BIGNUM_DIGIT_LENGTH - nbits)) |
         (BIGNUM_REF(a, size_a - 2) >> nbits));
    y = ((size_b >= size_a - 1 ? BIGNUM_REF(b, size_a - 2) >> nbits : 0) |
         (size_b >= size_a
              ? BIGNUM_REF(b, size_a - 1) << (BIGNUM_DIGIT_LENGTH - nbits)
              : 0));

    // inner loop of Lehmer's algorithm;
    A = 1;
    B = 0;
    C = 0;
    D = 1;
    for (k = 0;; k++) {
      if (y - C == 0)
        break;

      q = (x + (A - 1)) / (y - C);

      s = B + (q * D);
      t = x - (q * y);

      if (s > t)
        break;

      x = y;
      y = t;

      t = A + (q * C);

      A = D;
      B = C;
      C = s;
      D = t;
    }

    if (k == 0) {
      // no progress; do a Euclidean step
      if (size_b == 0) {
        return bignum_trim(a.untagged());
      }
      data_root<bignum> e(bignum_trim(a.untagged()), this);
      data_root<bignum> f(bignum_trim(b.untagged()), this);
      c.set_untagged(bignum_remainder(e.untagged(), f.untagged()));
      if (c.untagged() == BIGNUM_OUT_OF_BAND) {
        return c.untagged();
      }

      // copy 'b' to 'a'
      scan_a = BIGNUM_START_PTR(a);
      scan_b = BIGNUM_START_PTR(b);
      a_end = scan_a + size_a;
      b_end = scan_b + size_b;
      while (scan_b < b_end)
        *(scan_a++) = *(scan_b++);
      while (scan_a < a_end)
        *(scan_a++) = 0;
      size_a = size_b;

      // copy 'c' to 'b'
      scan_b = BIGNUM_START_PTR(b);
      scan_c = BIGNUM_START_PTR(c);
      size_c = BIGNUM_LENGTH(c);
      c_end = scan_c + size_c;
      while (scan_c < c_end)
        *(scan_b++) = *(scan_c++);
      while (scan_b < b_end)
        *(scan_b++) = 0;
      size_b = size_c;

      continue;
    }

    // a, b = A*b - B*a, D*a - C*b if k is odd
    // a, b = A*a - B*b, D*b - C*a if k is even

    scan_a = BIGNUM_START_PTR(a);
    scan_b = BIGNUM_START_PTR(b);
    scan_c = scan_a;
    scan_d = scan_b;
    a_end = scan_a + size_a;
    b_end = scan_b + size_b;
    s = 0;
    t = 0;
    if (k & 1) {
      while (scan_b < b_end) {
        s += (A * *scan_b) - (B * *scan_a);
        t += (D * *scan_a++) - (C * *scan_b++);
        *scan_c++ = (bignum_digit_type)(s & BIGNUM_DIGIT_MASK);
        *scan_d++ = (bignum_digit_type)(t & BIGNUM_DIGIT_MASK);
        s >>= BIGNUM_DIGIT_LENGTH;
        t >>= BIGNUM_DIGIT_LENGTH;
      }
      while (scan_a < a_end) {
        s -= (B * *scan_a);
        t += (D * *scan_a++);
        *scan_c++ = (bignum_digit_type)(s & BIGNUM_DIGIT_MASK);
        //*scan_d++ = (bignum_digit_type) (t & BIGNUM_DIGIT_MASK);
        s >>= BIGNUM_DIGIT_LENGTH;
        t >>= BIGNUM_DIGIT_LENGTH;
      }
    } else {
      while (scan_b < b_end) {
        s += (A * *scan_a) - (B * *scan_b);
        t += (D * *scan_b++) - (C * *scan_a++);
        *scan_c++ = (bignum_digit_type)(s & BIGNUM_DIGIT_MASK);
        *scan_d++ = (bignum_digit_type)(t & BIGNUM_DIGIT_MASK);
        s >>= BIGNUM_DIGIT_LENGTH;
        t >>= BIGNUM_DIGIT_LENGTH;
      }
      while (scan_a < a_end) {
        s += (A * *scan_a);
        t -= (C * *scan_a++);
        *scan_c++ = (bignum_digit_type)(s & BIGNUM_DIGIT_MASK);
        //*scan_d++ = (bignum_digit_type) (t & BIGNUM_DIGIT_MASK);
        s >>= BIGNUM_DIGIT_LENGTH;
        t >>= BIGNUM_DIGIT_LENGTH;
      }
    }
    BIGNUM_ASSERT(s == 0);
    BIGNUM_ASSERT(t == 0);

    // update size_a and size_b to remove any zeros at end
    while (size_a > 0 && *(--scan_a) == 0)
      size_a--;
    while (size_b > 0 && *(--scan_b) == 0)
      size_b--;

    BIGNUM_ASSERT(size_a >= size_b);
  }

  // a fits into a fixnum, so b must too
  fixnum xx = bignum_to_fixnum(a.untagged());
  fixnum yy = bignum_to_fixnum(b.untagged());
  fixnum tt;

  // usual Euclidean algorithm for longs
  while (yy != 0) {
    tt = yy;
    yy = xx % yy;
    xx = tt;
  }

  return fixnum_to_bignum(xx);
}
#endif

}